Elements of a Systematic Procedure for the Derivation of Macroscale Conservation Equations for Multiphase Flow in Porous Media

Abstract

The framework for development of conservation equations and constitutive relations describing the flow of two fluids in a porous medium from a macroscale perspective is provided. Theorems are employed that allow global integral equations to be localized at the porous medium scale. This is a more general approach than the traditional averaging of microscale point equations. Conservation equations for mass, momentum, energy and entropy for phases, interfaces, and common lines are obtained to provide a full description of two-phase flow in porous media. The entropy inequality is developed for the three-phase system. Because of interactions among the system components (i.e. phases, interfaces, and common lines), a single entropy inequality for the entire system, rather than inequalities for each phase, interface, and common line is employed in developing the constitutive theory. The macroscale internal energy is postulated to depend thermodynamically on the extensive properties of the system, and is then decomposed to provide energy dependences for each of the system components. These decomposed forms, along with the entropy inequality, lead to the conservation equations with constitutive approximations included. Final closure of the system of equations requires a variational analysis of the mechanical behavior of the subscale geometric structure. The resulting equations show that capillary pressure is a function of interphase area per unit volume as well as saturation. The equations currently used to model multiphase flow are shown to be very restricted forms of the more general equations.